In this previous post I described a block method for device noise parameter estimation. The method estimates device noise parameters by robust spatial averaging of local calibration image statistics. This post describes an improvement of this method and its application to light subframe background noise estimation. Background estimation of the noise in light subframes is useful for their quality evaluation and maximum likelihood integration.

The block method measures a sample statistic inside every non-overlapping *N* x *N* (typically 8 x 8) pixel block of an input image. A robust spatial average is then formed by computing the median of the measurements. The robustness of this process is further increased by computing the mean of results across multiple cycle-spins[1] and, if available, multiple images.

To check for block method systematic bias, I generated a stationary Gaussian noise image of size 1k x 1k pixels with mean 0 and standard deviation 1 and measured the median of the standard deviation of every non-overlapping 8 x 8 pixel block. I repeated this process for 128 similarly generated images and 8 cycle-spins of each image. The mean and standard deviation of the results equals 0.9948 and 0.0008, respectively. These results show that the block method slightly underestimates the standard deviation of stationary Gaussian noise by approximately 0.5%. I compensate for this small systematic bias by scaling all standard deviation estimates produced by the block method by the factor 1 / 0.9948.

The following results are for subframes captured with my Quantum Scientific Imaging 683wsg camera, binned 2 x 2 at -20° C. Light subframe examples have 2400 seconds of exposure.

I repeated the device noise estimation process described in the prior post to 32 pairs of dark subframes each with 2400 seconds of exposure and 8 cycle-spins of each pair. The subframes were captured in October 2012. The mean and standard deviation of the results are 14.07 DN and 0.01 DN, respectively.

I repeated the device gain estimation process described in the prior post to 32 quadruples of flat and flat dark subframes each with 24 seconds of exposure and 8 cycle-spines of each quadruple. The subframes were captured in October 2012. The mean and standard deviation of the results are 0.940 DN/e- and 0.001 DN/e-, respectively. These results are equal to the results provided by the detector manufacturer to within 0.5%.

To estimate the background noise of a light subframe, I first dark subtract and flat field the subframe to remove spatial noise and offset. Then I compute the median of the standard deviation of the blocks in the subframe whose mean is less than the *k*th quantile. This median forms the estimate of background noise. The mean of the *k*th/2 quantile block forms the estimate of background mean. What constitutes the background of a subframe is subjective. As a result the value of *k* is a free parameter. Its value is fixed for a set of subframes of a particular target to increase the relative consistency of the measurements of the set. Smaller values of *k* risk biases in the results due to undersampling the background. Larger values risk biases due to the inclusion of brighter, non-homogeneous target structures and intensity related variations of Poisson noise. For the examples below I used a value of *k* equal to 0.02. By experimentation I have found values in the range from 0.01 to 0.02 appropriate for subframes of the HII regions photographed in this gallery that contain limited amounts of homogeneous background.

The estimate of background noise includes contributions from both device noise and non-homogeneous target structures. To help distinguish between these two contributions, I compute a second estimate of background noise using a simple device noise profile. Device noise profile estimates are given by (*α**μ* + *σ*^2)^0.5, with the detector gain *α* and noise *σ* parameters set 0.940 DN and 14.1 DN as measured above, respectively, and *μ* equals the estimate of background mean as determined by the block method in DN units.

The image below shows the background blocks for 8 cycle-spins overlaid on an input subframe, a component the June, 2012 gallery photograph. Blue colored pixels are members of at least one background block with a standard deviation less than the median; red colored pixels are members of at least one background block with standard deviation greater than the median; purple colored pixels are members of at least one of both categories of background blocks on different cycle-spins. I applied a nonlinear transfer function to the subframe to improve its visibility.

The plot below shows a histogram of the standard deviation of the background blocks. The upper tail is primarily due to non-homogeneous target structures (e.g. dim stars) in the background. The median equals 20.2 DN.

The blue curve in the plot below shows the relationship between the block method background noise estimate as a function of quantile for the input subframe shown above. The magenta curve shows the corresponding device profile noise estimate. The chartreuse curve shows confusion, an estimate of noise due to non-homogenous background structures, defined by *β*^2 = *δ*^2 + *κ*^2, where *β*, *δ*, and *κ* are the block method estimate, the device profile estimate and confusion, respectively.

The plot below shows noise estimates for 19 subframes of the same target. The estimate variations are due to differences in observing conditions (e.g. atmospheric extinction and glow).

The image and plots below show an example of a subframe for a target that has negligible homogenous target structure in the background. This subframe is a component of the July, 2012 gallery photograph. The block method background noise estimate is significantly larger than the device profile noise estimate due to the presence of the non-homogeneous target structures.

The next example is a component of the August, 2012 gallery photograph.

The image and plots below an example of a dim target subframe, a component of the September, 2012 gallery photograph.

A final example, a component of the October, 2012 gallery photograph.

[1] Coifman et al., "Translation invariant de-noising", in *Lecture Notes in Statistics: Wavelets and Statistics*, 130:125-150, Springer Verlag, New York, 1995.

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